Minors in Boolean Structures and Matroids

Annals of Discrete Mathematics 14 (1982) 219-224

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MINORS I N BOOLEAN STRUCTURES AND MATROIDS L. GUIDOTTI and G. NICOLETTI I s t i t u t o d i Geometria U n i v e r s i t a d i Bologna

1 INTRODUCTION Forbidden minors are a useful device f r e q u e n t l y used i n Combinatorics and o t h e r branches o f Mathematics. This technique consists o f d e s c r i b i n g a c l a s s o f s t r u c tures as those which do n o t admit any given substructures, u s u a l l y named forbidden

.

minors I n t h i s way, the problem o f recognizing i f a p a r t i c u l a r s t r u c t u r e belongs t o a given class i s reduced t o checking whether i t contains some o f the excluded minors, o r not. I n t h i s paper, we introduce f i n i t e matroids as a class o f h e r e d i t a r y o r co-here! i t a r y systems, subjected t o some forbidden minors-condi t i o n s . Thus, we provide some new axiomatizations o f f i n i t e matroids, which seem t o be u s e f u l i n g i v i n g a more systematic approach t o the various cryptomorphic axiomatizations o f matroids. Hereditary systems and matroids a r e here defined i n terms o f subsets o f a f i n i t e boolean algebra: our language may cause some d i f f i c u l t i e s a t t h e beginning, b u t many conceptual s i m p l i f i c a t i o n s are made possible by i t . I n p a r t i c u l a r , i t can be e a s i l y seen t h a t the whole theory o f d u a l i t y i n matroid theory i s nothing b u t an instance o f t h e boolean d u a l i t y . 2

PRELIMINARIES AND NOTATION

I n t h i s paper, we w i l l deal w i t h f i n i t e boolean algebras. By t h e term " d u a l i t y " we w i l l r e f e r t o the boolean d u a l i t y . I f B denotes a f i n i t e boolean algebra, we w i l l denote by the same symbol B i t s underlying set. The l e a s t and the greatest element o f B w i l l be denoted by OB and lB, respectively, o r by 0 and 1 when no misunderstanding can a r i s e . We r e c a l l t h a t the h e i g h t o f an element x . o f a f i n i t e boolean algebra B, whose a t o m a r e ai,

i s defined as f o l l o w s : hg(x):=card{ai:

The height o f

ai(x).

B i s defined as h( B) := hB( 1 ) 219

220

L. Guidotti and G. Nicoletti

For any given x,ycB,

w i t h xsy, we w i l l s e t

Ix,Y(

:={zcB:xrz
Then, Ix,yJ i s a subalgebra o f B, w i t h respect t o the p a r t i a l order defined i n B. A descending s e t o f a f i n i t e boolean algebra B i s a ( p o s s i b l y empty) subset A o f

B such t h a t , f o r every x,ycB,

i f XLYEB, then xcA. Dually, an ascending s e t o f B i s

i f X ~ Y E B , then xcA.

a ( p o s s i b l y empty) subset A o f B such t h a t , f o r every x,ycB, For every subset A o f B, we s e t max(A):={xcA: x i s maximal i n A}, min(A):={xcA: x i s minimal i n A}, V (A) :={xcB:

t h e r e e x i s t s ycmax( A) ,xkyl,

A(A) :={xcB: t h e r e e x i s t s ycmin(A) ,xsy},

r( A) :={xEB:

XCAI.

A s t r u c t u r e , o r system, over a f i n i t e boolean algebra B i s a p a i r S:=(B,A),

r e A i s a subsev o f B. Two given systems Si:=(Bi,Ai),

i=1,2, w i l l k said

whe

to

be

isomorphic whenever t h e r e e x i s t s a boolean isomorphism f:B1-+B2 such t h a t f(A1)=A2. I f S1 and S2 are isomorphic, we w i l l s e t S1=S2.

dual

o f a s t r u c t u r e S:=(B,A) i s the s t r u c t u r e S":=(B",A), where 6" denotes The the dual o f the boolean algebra B. We have, obviously, Soo=S.I f S:=(B,A), we s e t

V(S) := (B ,V(A) , A(S) :=(B,A(A)) 9 r(S):=(B,r(A)). I t i s e a s i l y v e r i f i e d t h a t f o r any system S t h e f o l l o w i n g i d e n t i t i e s hold:

v(so)=(v(s))o, A(SO)=(A(S)1O ,

r(s")=(r(s))".

Now letS:=(B,A)

be a system over the boolean algebra B: a subsystem o r substru

c t u r e o r minor o f S w i l l IfSi:=(Bi,Ai),

be any system S(x,y) w i t h x,ycB,

xsy, and

-

S(X,Y) :=( I X,Y I ,An1 x,yl 1. i=1,2, a r e two given s t r u c t u r e s , we w i l l s e t S1<-S2 whenever

t h e r e e x i s t x,ycB2, xsy, such t h a t Sl~S2(xsy). Given a class C o f s t r u c t u r e s and a ( f i n i t e ) f a m i l y o f s t r u c t u r e s F, we w i l l say t h a t F i s a f a m i l y of forbidden ( o r excluded) minors f o r C i f , f o r every s t r u c t u r e T we have: T i s i n C i f and only i f s f o r every S b , S / T . I t i s worthwile t o remark t h a t d u a l i t y and t h e operator

r

preserve minors, and

consequently forbidden minors:

Proposition 1 Let S i : = ( B . , A . ) , i=1,2 be two systems: then S j sS2 if and only if 2 2 Slo5Szo and SIs S2 if and only if r(Sl)
22 1

Minors in Boolean structures and matroids 3

FORBIDDEN MINORS FOR HEREDITARY SYSTEMS

A h e r e d i t a r y system ( o r independence system) i s a system S=(B,A) where A i s a descending f a m i l y o f B; co-hereditary systems a r e defined d u a l l y . The systems (B,0),

(B,{OBI),

(B,B)

a r e h e r e d i t a r y s t r u c t u r e s which w i l l be c a l l e d

the degenerate, t r i v i a l and d i s c r e t e h e r e d i t a r y systems, r e s p e c t i v e l y . Dually, (B,0), (B,{lBI), (B,B) a r e co-hereditary systems, which w i l l be c a l l e d the

erate, trivial

a

and d i s c r e t e co-heredi t a r y sistems, r e s p e c t i v e l y .

We w i l l denote by P and Po the t r i v i a l co-hereditary s t r u c t u r e and the t r i v i a l h e r e d i t a r y s t r u c t u r e over a boolean algebra o f h e i g h t 1, r e s p e c t i v e l y .

P

P

PO

Figure 1

P i s the unique system over a boolean algebra o f h e i g h t 1 which i s P? Furthermore, P i s t h e forbidden minor f o r t h e c l a s s o f h e r e d i t a r y systems, and d u a l l y f o r P: as s t a t e d i n t h e f o l l o w i n g

We remark t h a t

n o t a h e r e d i t a r y system, and d u a l l y f o r

proposition:

Proposition 2 Let S:=(B,A) be a system over a f i n i t e boolean aZgebra B: then S is a hereditary system if and onZy if [email protected], S is a co-hereditary system if and only if Po+$. 4

FORBIDDEN MINORS FOR MATROIDS As i s w e l l known, f i n i t e matroids can be defined i n various cryptomorphic ways.

Here, we a r e i n t e r e s t e d i n regarding them as h e r e d i t a r y ( o r co-hereditary) s t r u c tures, subjected t o some a d d i t i o n a l conditions. More p r e c i s e l y , an independent matroid ( i - m a t r o i d f o r s h o r t ) i s a h e r e d i t a r y system M=(B,I), where I s a t i s f i e s the f o l l o w i n g "augmentation axiom" : f o r every x,ycI, hB(x)hB(y), t h e r e e x i s t s ZES such t h a t x>z and xvy>zVy. The elements o f S w i l l be s a i d t o be the spanning elements o f M.The degenerate coh e r e d i t a r y systems wi.11 be c a l l e d the degenerate s-matroids.

L. Guidotti and G. Nicoletti

222

A dependence matroid ( o r d-matroid ) i s a co-hereditary system M=(B,D), where D s a t i s f i e s t h e f o l l o w i n g axiom: f o r every dl,d2ED, i f dlAd2 C D then t h e r e e x i s t s d3ED such t h a t dgdlVd2 and he( d3)=hB(dlvd2)-l. The elements o f D w i l l be c a l l e d t h e dependent elements o f M.The d i s c r e t e co-hered it a r y systems w i 11 be regarded as degenerate d-matroids. A non-spanning matroid ( o r n-matroid ) i s a h e r e d i t a r y system M=(B,N) where N s a t i s f i e s t h e f o l l o w i n g axiom: f o r every nl,n2EN,'

i f nlVn2LN,

then t h e r e e x i s t s n3EN such t h a t n p l A n 2

hB(n3)=hB(nlAn2)t1. The elements o f N w i l l be c a l l e d t h e non-spanning elements o f

M.

and

The d i s c r e t e he

r e d i t a r y systems w i 11 be s a i d t o be degenerate s-matroids. The l i n k s between t h e classes o f s t r u c t u r e s now defined a r e described by t h e f o l 1owi ng proposi ti ons :

Proposition 3 Let S:=IB,A) be a system over the f i n i t e boolean algebra B.Then:

i) ii) iii) iv) v) vi) Proof

S is a d-matroid S is an i-matroid S is an a-matroid S is an n-matroid S is an s-matroid S is an a-matroid See [4],

[6]

.

i f and only i f rlS) is an i-matroid;

if and only i f V($)

is an s-matroid;

i f and only i f r(S) is an n-matroid;

i f and only if r(S) is an s-matroid;

if and only i f ii(S) is an i-matroid; i f and only i f r l S ) is an d-matroid.

Proposition 4 Let S:=(B,AI

be a system over a f i n i t e boolean algebra B. Then:

il S is a d-matroid if and only if So is an n-matroid; ii) S is an i-matroid i f and only i f So is an s-matroid. Proof I t f o l l o w s from d u a l i t y between axioms d e f i n i n g d-matroids and n-matroids, i-matroids and s-matroids, r e s p e c t i v e l y . Now,let M1 be a d-matroid, and s e t M2:=r(M1), M3:=v(M2), M4:=r(M3); then we w i l l r e f e r t o Mi, i=l,2,3,4 as t h e cryptomorphic representation o f t h e same Et r o i d M. Then M:, i=1,2,3,4 a r e cryptomorphic representations o f a second matroid o f M. M'which w i l l be c a l l e d the I t i s easy t o check t h a t a l l t h e h e r e d i t a r y s t r u c t u r e s over boolean algebras o f h e i g h t 1 o r 2 are i-matroids, and dually, a l l the co-hereditary s t r u c t u r e s 2 ver boolean algebras o f h e i g h t 1 o r 2 a r e s-matroids. Now l e t B be a boolean algebra o f h e i g h t 3 whose atoms a r e ai, i=1,2,3. The system Q:=(B,A) defined by t h e f a m i l y A=IOB,al ,a2,a3,alva31 i s a hereditary s t r u g under boolean isomorphisms - the unique h e r e d i t a r y system ture; moreover, Q i s

dual

-

223

Minors in Boolean structures and matroids over B which i s n o t an i - m a t r o i d . Dually, t h e s t r u c t u r e o_"=v(O_) i s phisms

-

-

under isomor

t h e unique c o h e r e d i t a r y system which i s n o t a s-matroid.

2

p" Figure 2

Furthermore, we have: Theorem 1 Let S : = ( B , A ) be a hereditary system over the f i n i t e algebra B; then S

i s an i-matroid if and only i f Q@. Dually, l e t S:=(B,AI be a oo-hereditary system over B; then S i s a s-matroid i f a n d only i f p"$S. P r o o f L e t t h e h e r e d i t a r y system S be an i - m a t r o i d , and l e t aEB, biEB, bi>a, hB(bi)=hB(a)+l, i=l,2,3, and s e t c1=b2vb3, c2=blvb3, c3=blvb2, d=clvc2=c2vc3= =c1vc3 * L e t us suppose b,cA and cleA: then,by t h e augmentation axiom, t h e r e e x i s t s ZEA such t h a t blx and Z A Y = ( V ~ V X ) A Ay=(v1Ay) > x ~ y ,showing t h a t t h e augmentation p r o p e r t y holds. By t h e previous theorem, we can say t h a t

{P,2>i s

f o r t h e c l a s s o f i - m a t r o i d s , and d u a l l y , {P",o,") f o r t h e c l a s s o f s-matroids.

a f a m i l y o f f o r b i d d e n minors

i s a f a m i l y o f f o r b i d d e n minors

Ro=r(v),

L e t us now consider t h e systems R=r(Q) and p i c t u r e d i n F i g u r e 3. I t i s easy t o see t h a t R i s t h e unique c o - h e r e d i t a r y system o v e r a boolean algebra o f h e i g h t 3 which i s n o t a d-matroid, and d u a l l y Ro i s t h e unique h e r e d i t a r y s y z

L. Guidotti and C.Nicoletti

224

tem over a boolean algebra o f h e i g h t 3 which i s n o t an n-matroid.

R

R0

Figure 3 Applying Proposition 1 we get immediately:

Let S:=(A,BI be a co-hereditary system over the f i n i t e algebra B; Theorem 2 then S is a d-matroid if and only i f &@. Dually, l e t S:=(A,BI be a hereditary 8yc tem, then S is an n-matroid if and only if RqS. As a consequence o f the l a s t theorem, {Po,R} and {P,Ro} are f a m i l i e s o f f o r b i d den minors f o r the classes o f d-matroids and n-matroids, r e s p e c t i v e l y . REFERENCES

[l]Brylawski ,T.H. ,Kelly,D.G. and Lucas,T.D., Vatroids and combinatorial Ceomg t r i e s , Lecture Note Series, U n i v e r s i t y o f North Carolina, Chapel Hi11.(1974). [2] Crap0,H.H.

and Rota,G.C.,

t o r i a l Geometries, I1.I.T.

On the foundations o f Combinatorial Theory: Combin2 Press Cambridge (1970).

[3] Las Vergnas,M. ,Sur l a d u a l i t e en t h e o r i e des matroides,C.R.Acad.Sci.(Paris)

13

(1970) 804-806. [4] Pezzoli ,L.,

Sistemi d i indipendenza modulari, B.U.M.I.

( 5 ) 18-8 (1981)167-180.

151 Tutte,W.I.,

I n t r o d u c t i o n t o t h e theory o f matroids (American Elsevier, New

York, 1970). [6] Welsh,D.J.A.,

F a t r o i d theory (Academic Press, London, 1976).